Philosophy 1100
Class #9
Title:
Critical Reasoning
Instructor:
Paul Dickey
E-mail Address: pdickey2@mccneb.edu
Website: http://mockingbird.creighton.edu/NCW/dickey.htm
Today:
Submit Final Essay
Turn in Exercise 9-14
Return Portfolios
Discuss Chapter 10
Next Week:
No class
Final Exam (Take-home)
1
Final Exam will be posted on Quia by Noon, Tuesday, 11/17.
Most questions are multiple choice but that there will be some
questions that will require drawing & free form text.
Complete the exam in one of the following ways:
•
Download and print the document from the Quia site.
Answer all questions on the hardcopy exam you print.
After you are done, scan the exam as a jpg, pdf, doc, or
docx file and email it to me at pdickey2@mccneb.edu;
or
•
Download and complete the exam by editing it within
Microsoft Word and email it to me as a doc or docs file
to pdickey2@mccneb.edu.
E-mailed exams must be received no later than 6:00
P.M. on 11/23. Please Note: For every 8 hours (or
partial) the exam is late, a full grade will be reduced.
NO EXCEPTIONS.
2
As before, you may consult your notes, your
laminated cards, or textbook while taking the exam,
but not other students.
On your exam somewhere, please discuss which 3
(at least 3, more if you like) questions you found the
most difficult and what other answers did you find
“plausible.” This will help me to evaluate the exam
for the future. Your exam will not be considered
complete without this feedback.
Congratulations, you have completed the course.
You should be proud. After you have finished the
exam and e-mailed it to me, take the evening off
and do something nice for yourself. I have enjoyed
the class and hope you have also.
3
Chapter Nine
Deductive Arguments:
Categorical Logic
Class Workshop
Exercise 9-14
Chapter Ten
Deductive Arguments:
Truth-Functional Logic
•
Truth Functional logic is important because it gives
us a consistent tool to determine whether certain
statements are true or false based on the truth or
falsity of other statements.
•
A sentence is truth-functional if whether it is true or not
depends entirely on whether or not partial sentences
are true or false.
•
For example, the sentence "Apples are fruits and
carrots are vegetables" is truth-functional since it is true
just in case each of its sub-sentences "apples are
fruits" and "carrots are vegetables" is true, and it is
false otherwise.
•
Note that not all sentences of a natural language, such
as English, are truth-functional, e.g. Mary knows that
the Green Bay Packers won the Super Bowl.
Truth Functional Logic: The Basics
•
Please note that while studying Categorical Logic, we
used uppercase letters (or variables) to represent
classes about which we made claims.
•
In truth-functional logic, we use uppercase letters
(variables) to stand for claims themselves.
•
In truth-functional logic, any given claim P is true or
false.
•
Thus, the simplest truth table form is:
P
_
T
F
Truth Functional Logic: The
Basics
•
Perhaps the simplest truth table operation is negation:
P
~P
T
F
F
T
Truth Functional Logic: The
Basics
•
•
Now, to add a second claim, to account for all truthfunctional possibilities our representation must state:
P
Q
T
T
F
F
T
F
T
F
And the operation of conjunction is represented by:
P
Q
P & Q
T
T
F
F
T
F
T
F
T
F
F
F
Truth Functional Logic: The
Basics
•
•
The operation of disjunction is represented by:
P
Q
T
T
F
F
T
F
T
F
P V Q
T
T
T
F
The operation of the conditional is represented by:
P
Q
T
T
F
F
T
F
T
F
P -> Q
T
F
T
T
Using Truth Tables To Test Validity
• Now, consider the following argument:
Premise: If Paula goes to work, then Quincy and Rogers
will get a day off.
Conclusion: If Paula goes to work and Quincy gets a day
off, then Rogers will get a day off.
• We symbolize the conclusion as (P & Q) -> R
• Thus, the argument is:
P -> (Q & R)
(P & Q) -> R
• Is this a valid argument?
Using Truth Tables To Test Validity
• Is this a valid argument? We can determine whether or
not it is by constructing a truth table that presents the
premise(s) and conclusion.
• In this case, to do so we add to the previous truth table the
necessary columns to represent the conclusion.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
Q&R
T
F
F
F
T
F
F
F
P -> (Q & R)
T
F
F
F
T
T
T
T
P&Q
T
T
F
F
F
F
F
F
(P & Q) -> R
T
F
T
T
T
T
T
T
• Now, remembering the definition of a deductive argument,
we look for a row in the table in which the premise(s) is true
but the conclusion is not true. If we find one, the argument is
invalid. If there is none, then the argument is valid.
Using Truth Tables To Test Validity
We can determine whether or not a deductive argument is
valid or invalid by constructing a truth table that presents the
premise(s) and conclusion.
A deductive argument is valid when if the premises are true,
the conclusion has to be true. Or in other words, an argument
is valid if there are no possible states or conditions in which
the premises are true and the conclusion is false.
And, of course, a truth table represents all the possible states
or conditions of the claims.
Thus, an argument is valid when there are NO rows of the
truth table in which the premise(s) are true and the conclusion
is not true. If there is even one, the argument is invalid.
Consider the
following
argument:
P -> Q
~P
_________
~Q
1. Construct the appropriate truth table to include all
possible t-f scenarios for all variables in the
argument.
If there are x (e.g. 2) variables, note that there with
always be x (so in this case, 2) columns in the truth
table at this point and there will be 2**x (or 2 to the x
power) number of rows (in this case, 4).
2. Add a column to the truth table to
express the first premise based on
the truth tables for the basic
operations.
You may have to do this in multiple steps.
P1
P
Q
P->Q
T
T
T
T
F
F
F
T
T
F
F
T
P
Q
T
T
T
F
F
T
F
F
Consider the
following
argument:
P -> Q
~P
_________
~Q
3. For each remaining premise
(there more may be more than one)
add a column to the truth table to
express the premise based on the
truth tables for the basic operations.
4. Add a column to the truth
table to express the
conclusion based on the
truth tables for the basic
operations.
P1
P2
~P
P
Q
P->Q
T
T
T
F
T
F
F
F
F
T
T
T
F
F
T
T
P1
P2
C
~P
~Q
P
Q
P->Q
T
T
T
F
F
T
F
F
F
T
F
T
T
T
F
F
F
T
T
T
You may have to do steps #3 and #4 also in multiple steps.
Consider the following argument:
P -> Q
~P
______
~Q
5. Ask yourself “Are there any rows in
the truth table that I have just
created in which all premises are
true and the conclusion is false?”
P1
P2
C
~P
~Q
P
Q
P->Q
T
T
T
F
F
T
F
F
F
T
F T
T
T
F
F
T
T
T
6. If the answer is yes, then write “invalid.”
If the answer is no, write “valid.”
Invalid
F
Laziness is the Mother of Invention
However there is often an easier way to demonstrate
validity with truth tables. It is called the short truthtable method.
The basic principle of this method simply is to look for a
row that makes the argument invalid. As soon as you
find one, you are done. If you exhaust all
opportunities and can’t, then the argument is valid.
Consider the argument:
P -> Q
~Q -> R
~P -> R
The argument could be invalid only if the conclusion is
false while the premises are true.
P Q R
F T F
Thus, the argument is invalid.
Now, consider the argument:
(P v Q) -> R
S -> Q
S -> R
The argument could be invalid only if the conclusion is
false while the premises are true.
To make the conclusion false -P
Q
R
F
S
T
To make the second premise true -P
Q
T
R
F
S
T
But there is no way now to make the first premise
true, so the argument is valid.
Exercises 10-6
1.
K -> (L & G)
M -> (J & K)
B&M
B&G
To make the third premise true –
B
T
M
T
But to make premise 2 true
B
T
M
T
J K
T T
But to make premise 1 true,
B
T
M
T
J K L G
T T T T
But there is no way now to make the conclusion false, so the
argument is valid.
Class Workshop (Short Method)
Translation Exercise:
If Scarlet is guilty of the crime, then Ms. White
must have left the back door unlocked and the
colonel must have retired before ten o’clock.
However, either Ms. White did not leave the back
door unlocked, or the colonel did not retire before
ten. Therefore, Scarlet is not guilty of the crime.
Now, is is this valid? Look at pages 301
& 302 for the long proof. Wow! Now,
let’s be “lazy”…
Deductive
Arguments:
Rules of Induction
Deduction: Group 1 Rules
The basic valid argument patterns of deductive logic
(If doubted, all the rules we discuss below can be confirmed by the
truth-table method) is another method to prove a deductive
argument (that is, to show that it is valid).
•
Modus Ponens (MP)
P -> Q
P____
Q
-- Affirming the antecedent
P
T
T
F
F
Q
T
F
T
F
P->Q
T
F
T
T
Deduction: Group 1 Rules
2.
Modus Tollens (MT)
P -> Q
~Q____
~P
-- Denying the consequent
Deduction: Group 1 Rules
Okay, now that we have two rules to play
with, let’s stop for a minute and see how we
prove an argument valid using the rules.
1.
2.
3.
4.
5.
(P & Q) -> R
S
S -> ~R
~R
~ (P & Q)
/ .’. ~ ( P&Q)
2,3, MP
1,4, MT
3.
Chain Argument
(CA)
P -> Q
Q -> R____
P -> R
4.
Disjunctive Argument
PvQ
~P
Q
5.
Simplification (SIM)
P&Q
P
6.
PvQ
~Q__
P
P&Q
Q
Conjunction (CONJ)
P
Q__
P&Q
(DA)
7.
Addition
P
PvQ
8.
(ADD)
Q
PvQ
Constructive Dilemma
(CD)
P –> Q
R -> S
P v R
Q v S
9.
Destructive Dilemma
P –> Q
R -> S
~Q v ~S
~P v ~R
(DD)
Exercises 9-10
#1
1.
2.
3.
R -> P
Q -> R
Q -> P
/ .’. Q -> P
1,2, CA
#2
1.
2.
3.
4.
P -> S
PvQ
Q -> R
SvR
/ .’. S v R
1,2,3, CD
#10
1.
2.
3.
4.
5.
6.
7.
(T v M) -> ~Q
(P -> Q) & (R-> S)
T
TvM
~Q
P -> Q
~P
/ .’. ~P
3, ADD
1, 4, MP
2, SIMP
5, 6, MT